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C. R. Acad. Sci. Paris, Ser. I 354 (2016) 517–521
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Algebraic geometry
Genus one enumerative invariants in del-Pezzo surfaces
with a fixed complex structure
Invariants énumératifs de genre un avec une structure complexe fixée pour
des surfaces de del Pezzo
Indranil Biswas a , Ritwik Mukherjee a , Varun Thakre b
a
b
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India
Department of Mathematics, Harish Chandra Research Institute, Allahabad 211019, India
a r t i c l e
i n f o
Article history:
Received 28 September 2015
Accepted after revision 24 February 2016
Available online 24 March 2016
Presented by Claire Voisin
a b s t r a c t
We obtain a formula for the number of genus one curves with a fixed complex structure of
a given degree on a del-Pezzo surface that pass through an appropriate number of generic
points of the surface. This enumerative problem is expressed as the difference between the
symplectic invariant and an intersection number on the moduli space of rational curves.
© 2016 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.
r é s u m é
Nous obtenons une formule pour le nombre de courbes de genre un avec une structure
complexe fixée, de degré donné, et passant par un nombre approprié de points génériques
de la surface. La solution est exprimée comme la différence entre l’invariant symplectique
et un nombre d’intersection sur l’espace de modules de courbes rationnelles.
© 2016 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.
1. Introduction
Enumerative Geometry of rational curves in P2C is a classical question in algebraic geometry. A natural generalization is
to ask how many elliptic curves, with a fixed j-invariant, are there of a given degree that pass through the right number
of generic points. In [8] and [4], using methods of algebraic and symplectic geometry respectively, Pandharipande and Ionel
obtain a formula for the number of degree d genus one curves with a fixed complex structure in P2C that pass through
3d − 1 generic points. In this paper, we extend their result to del-Pezzo surfaces.
Let X be a complex del-Pezzo surface and β ∈ H 2 ( X , Z) a given homology class. Let n0,β denote the number of rational
curves of degree β in X that pass through δβ generic points, where δβ := c 1 (TX ), β − 1. We prove the following.
E-mail addresses: indranil@math.tifr.res.in (I. Biswas), ritwikm@math.tifr.res.in (R. Mukherjee), varunthakre@hri.res.in (V. Thakre).
http://dx.doi.org/10.1016/j.crma.2016.02.009
1631-073X/© 2016 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.
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I. Biswas et al. / C. R. Acad. Sci. Paris, Ser. I 354 (2016) 517–521
j
Theorem 1.1. Let X and β be as above. Let n1,β denote the number of elliptic curves with fixed j invariant of degree β in X that pass
through δβ generic points. Then
2g β
j
n1,β =
|Aut(1 , j )|
n0,β
where
g β :=
β · β − c 1 (TX ) · β + 2
2
,
|Aut(1 , j )| denotes the number of automorphisms of a genus one Riemann surface with fixed j invariant that fixes a point and “·”
denotes topological intersection.
Note that g β in Theorem 1.1 coincides with the genus of a smooth degree β curve on X . The numbers n0,β are computed
j
in [6, p. 29] and [2, Theorem 3.6] using a recursive formula. When X := P2 , our formula for n1,β is consistent with the
formula of Pandharipande and Ionel in [8] and [4]. In [6], the authors actually give a formula to compute the genus 0
Gromov–Witten invariants of the del-Pezzo surfaces, which a priori need not be enumerative. It is shown in [2, page 63,
last paragraph] that the numbers obtained in [6] are actually equal to n0,β .
The results of Pandharipande and Ionel generalize the result of P. Aluffi; in [1], he computes the number of genus one
cubics with a fixed complex structure in P2 through 8 generic points.
The problem of enumerating elliptic curves with a fixed j-invariant has also been studied by tropical geometers. In [5],
Kerber and Markwig compute the number of tropical elliptic curves in P2 with a fixed j-invariant. Combined with the
correspondence theorem [7, Theorem A], one can conclude that the number computed is indeed the same as the number of
plane elliptic curves with a fixed j-invariant. Currently, this question is also being studied for other surfaces. In [7], Len and
Ranganathan obtain a formula for the number of elliptic curves with a fixed j-invariant of a given degree for Hirzebruch
surfaces, using methods from tropical geometry.
2. Enumerative versus symplectic invariant
j
We now explain the basic idea to compute n1,β . Let ( X , J , ω) be a compact semi-positive symplectic manifold, with a
compatible almost complex structure J of dimension 2m and β ∈ H 2 ( X , Z) be a homology class. Let k be a nonnegative
integer such that k + 2g ≥ 3. Let α1 , · · · , αk and γ1 , · · · , γl be integral homology classes in H ∗ ( X , Z) such that
k
2m − deg(αi ) +
l
i =1
(2m − 2 − deg(γ j )) = 2m(1 − g ) + 2c 1 (TX ), β .
(2.1)
j =1
Fix pseudocycles A i , 1 ≤ i ≤ k, and B j , 1 ≤ j ≤ l, on X representing the homology classes αi and γ j . Fix a compact
Riemann surface g of genus g; its complex structure will be denoted by j. Define
M g ,k ( X , β; α1 , · · · , αk ; γ1 , · · · , γk ) := {(u , y 1 , · · · , yk ) ∈ C ∞ ( g , X ) × X k | u ∗ [ g ] = β,
ν, j
∂ j , J u = ν , u ( y i ) ∈ A i ∀ i = 1, · · · , k, Im(u ) ∩ B j = ∅ ∀ j = 1, · · · , l} ,
where ν : g × X −→ T ∗ g ⊗ TX is a generic smooth perturbation and
∂ j , J u :=
1
du + J ◦ du ◦ j .
2
The symplectic invariant (or the Ruan–Tian invariant) is defined to be the signed cardinality of the above set, i.e.,
ν, j
RT g ,β (α1 , · · · , αk ; γ1 , · · · , γl ) := ±|M g ,k ( X , β; α1 , · · · , αk ; γ1 , · · · , γk )| .
When k = 0, we denote the invariant as
RT g ,β (∅; γ1 , · · · , γl ).
Furthermore, when γ1 , . . . , γl all denote the class of a point, then we abbreviate the invariant as RT g ,β . Similarly, when l = 0
we denote the invariant as
RT g ,β (α1 , · · · , αk ; ∅).
If (2.1) is not satisfied, then we formally define the invariant to be zero.
j
A natural question is to ask whether the symplectic invariant RT g ,β is equal to the enumerative invariant n g ,β . For P2 ,
and more generally for del-Pezzo surfaces, the genus zero symplectic invariant is equal to the enumerative invariant [9,
page 267]. However, even for P2 , the genus one symplectic invariant is not enumerative. In general, the following fact is
true ([11, Theorem 1.1]),
j
RT1,β = |Aut(1 , j )|n1,β + CR ,
(2.2)
I. Biswas et al. / C. R. Acad. Sci. Paris, Ser. I 354 (2016) 517–521
519
where CR denotes a correction term. Let us explain what this term means. First we note that the factor of |Aut (1 , j )| is
ν, j
there because we do not mod out by automorphisms in the definition of M g ,k . Hence, if u : (1 , j ) −→ X is a solution to
the ∂ -equation and the complex structure on X is genus one regular, then there will be |Aut(1 , j )| new solutions close to
u to the perturbed ∂ -equation. Next, we note that as ν → 0, a sequence of ( J , ν )-holomorphic maps can also converge to a
bubble tree whose base (the torus) is a constant (ghost) map [4, page 2]. These maps will also contribute to the computation
of RT1,β invariant. This extra contribution is defined to be the correction term CR.
We now explain how to compute the correction term. Let M0,n ( X , β) denote the moduli space of rational degree β
curves on X that represent the class β ∈ H 2 ( X , Z) and are equipped with n ordered marked points, modulo equivalence.
In other words,
M0,n ( X , β) := {(u , y 1 , · · · , yn ) ∈ C ∞ (P1 , X ) × (P1 )n | ∂ u = 0, u ∗ [P1 ] = β}/PSL(2, C) ,
with PSL(2, C) acting diagonally on P1 × (P1 )n . Let M0,n ( X , β) denote the stable map compactification of M0,n ( X , β).
Let us now focus on M0,1 ( X , β), the moduli space of curves with one marked point. Let H be the divisor in
M0,1 ( X , β) corresponding to the extra condition that the curve passes through a given point. Let L −→ M0,1 ( X , β) and
ev : M0,1 ( X , β) −→ X be the universal tangent bundle and the evaluation map at the marked point. Following the same
argument as in [4, Lemma 1.23], we conclude that the bundle ev∗ TX −→ M0,1 ( X , β) ∩ Hδβ admits a nowhere vanishing
section ν . This is because the rank of ev∗ TX is two, while the dimension of the variety M0,1 ( X , β) ∩ Hδβ is one. Hence
ev∗ TX −→ M0,1 ( X , β) ∩ Hδβ admits a trivial sub bundle spanned by ν , which we denote as Cν . When X := P2 , it is
shown in [4, Lemma 1.25], that the correction term is given by
CR = c 1 (L∗ ⊗ ev∗ TX /Cν ), [M0,1 ( X , β)] ∩ Hδβ .
(2.3)
A more detailed justification of (2.3) is given in [11], by using the results of [12]. Furthermore, the gluing construction in
[12] is valid in general for Kähler manifolds [12, page 8]. Hence, we conclude that (2.3) holds for del-Pezzo surfaces as well.
Zinger also pointed out this fact to the second author of this paper in a personal communication ([13]).
In the next section we will obtain a formula for c 1 (L∗ ) and compute the right-hand side of (2.3). The left-hand side of
j
(2.2) is computed using the formula given in [9]. Hence, we obtain n1,β .
3. Computation of the correction term
We will now give a self-contained proof of obtaining a formula for c 1 (L∗ ) and hence computing the correction term.
Alternatively, one can also compute the Chern classes by using the dilaton equation and the divisor equation as given in [3,
Section 26.3].
Lemma 3.1. On M0,1 ( X , β), the following equality of divisors holds:
c 1 (L∗ ) =
(x1 · x1 )H − 2(β · x1 )ev∗ (x1 ) +
(β · x1 )2
1
Bβ1 ,β2 (β2 · x1 )2 ,
(3.1)
β1 +β2 =β,
β1 ,β2 =0
where H is the locus satisfying the extra condition that the curve passes through a given point, Bβ1,β2 denotes the boundary stratum
corresponding to the splitting into a degree β1 curve and degree β2 curve with the last marked point lying on the degree β1 component
and xi := c i (TX ).
Proof. The proof is similar to the one given in [4]. Let μ1 , μ2 ∈ X be two generic pseudocycles in X that represent the
be a cover of M0,1 ( X , β) with two additional marked points with the last two marked
class Poincaré dual to x1 . Let M
points lying on μ1 and μ2 respectively. More precisely,
:= ev−1 (μ1 ) ∩ ev−1 (μ2 ) ⊂ M0,3 ( X , β) ,
M
2
3
where ev2 and ev3 denote the evaluation maps at the second and third marked points respectively. Note that the projection
−→ M0,1 ( X , β) that forgets the last two marked points is a (β · x1 )2 -to-one map. We now construct a meromorphic
π:M
section
−→ π∗ L∗
φ:M
given by
φ([u , y 1 ; y 2 , y 3 ]) :=
( y 2 − y 3 )dy 1
.
( y 1 − y 2 )( y 1 − y 3 )
(3.2)
The right-hand side of (3.2) involves an abuse of notation: it is to be interpreted in an affine coordinate chart and then
extended as a meromorphic section on the whole of P1 . Note that on (P1 )3 , the holomorphic line bundle
η := q∗1 K P1 ⊗ O(P1 )3 (
12 +
13 −
23 )
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is trivial, where q1 : (P1 )3 −→ P1 is the projection to the first factor and jk ⊂ (P1 )3 is the divisor consisting of all points
(zi , z2 , z3 ) such that z j = zk . The diagonal action of PSL(2, C) on (P1 )3 lifts to η preserving its trivialization. The section φ
in (3.2) is given by this trivialization of η .
Since c 1 (π∗ L∗ ) is the zero divisor minus the pole divisor of φ , we gather that
c 1 (π∗ L∗ ) = { y 2 = y 3 } − { y 1 = y 2 } − { y 1 = y 3 } .
When projected down to M0,1 ( X , β), the divisor { y 2 = y 3 } becomes
(x1 · x1 )H + (β2 · x1 )2 Bβ1 ,β2 ,
is a (β · x1 )2 -to-one cover of M0,1 ( X , β),
while both the divisors { y 1 = y 2 } and { y 1 = y 3 } become (β · x1 )ev∗ (x1 ). Since M
we obtain (3.1). 2
Using Lemma 3.1, we conclude that
c 1 (L∗ ), [M0,1 ( X , β)] ∩ Hδβ = −2n0,β .
(3.3)
( X , β) ∩ Hδβ +1 is zero. This is because the number of rational curves through
To see why this is so, we first note that M0,1
δβ + 1 generic points is zero. Next, we note that M0,1 ( X , β) ∩ Hδβ ∩ Bβ1 ,β2 is also zero. This is because the number of β
curves which pass through δβ points cannot split into a degree β1 curve and a degree β2 curve. This is because such a split
curve will pass through δβ1 + δβ2 points, which is one less than δβ . So a split curve cannot pass through δβ generic points.
Finally we note that for any homology class μ ∈ H 2 ( X , Z), the following is true
[M0,1 ( X , β)] ∩ Hδβ ∩ ev∗ [μ] = n0,β (β · μ).
(3.4)
To see why this is so, we note that the left-hand side of (3.4) counts the number of degree β rational curves through δβ
points and one marked point, such that the marked point lies on some cycle representing the class β . There are β · μ choices
for that marked point to lie, which gives us the right hand side of (3.4). These three facts give us (3.3). Note that, when we
say ev∗ [μ], we mean the pullback of the cohomology class Poincaré dual to μ (inside X ). Using (3.4) (with μ := c 1 (TX )),
we conclude that
c 1 (ev∗ TX ), [M0,1 ( X , β)] ∩ Hδβ = β · c 1 (TX ) n0,β .
(3.5)
From (3.3), (3.5) and (2.3) it follows that
CR = β · c 1 (TX ) − 2 n0,β .
(3.6)
4. Computation of the symplectic invariant
We now compute the symplectic invariant RT1,β := RT1,β (∅; p 1 , . . . , p δβ ) using the formula [9, page 263, (1.2)]. Let
e 1 , e 2 , · · · , ek be a basis for H ∗ ( X , Z). Let
g i j := e i · e j
g i j :=
and
g −1
ij
.
If the degrees of e i and e j do not add up to be the dimension of X then define g i j to be zero. Using [9, page 263, (1.2)] we
conclude that
RT1,β (∅; p 1 , . . . , p δβ ) =
g i j RT0,β (e i , e j ; p 1 , . . . , p δβ ) =
i, j
g i j n0,β (β · e i )(β · e j )
i, j
= (β · β)n0,β .
(4.1)
The last equality follows by writing β in the given basis e i and using the definition of g i j ; the second equality follows from
the same we justify (3.4). Equations (4.1), (3.6) and (2.2) give us the formula of Theorem 1.1.
5. Regularity of the complex structure for del-Pezzo surfaces
We now show that the complex structure on the del-Pezzo surfaces is genus one regular for immersion. In the statement
of Theorem 1.1, the curve u passes through δβ generic points. Hence the curve is going to be an immersion and hence it
suffices to prove regularity for immersions.
Lemma 5.1. Let X be P2 blown up at k points and (1 , j ) a compact genus 1 Riemann surface with a complex structure j. Let
u : 1 −→ X be a holomorphic map representing the class β := dL − m1 E 1 − . . . − mk E k ∈ H 2 ( X , Z), where L and E i denote the
class of a line and the exceptional divisors respectively. Then D u , the linearization of the ∂ j , J at u is surjective, provided d > 0 and u is
an immersion. In particular, the complex structure on the del-Pezzo surface is genus 1 regular for immersions.
I. Biswas et al. / C. R. Acad. Sci. Paris, Ser. I 354 (2016) 517–521
521
Proof. We first note that if L is a holomorphic line bundle on (1 , j ) of positive degree, then the cup product
H 0 (1 , L) ⊗ H 1 (1 , L∗ ) −→ H 1 (1 , L) ⊗ L∗ ) = C
is nondegenerate. Indeed, this coincides with the Serre duality pairing because the canonical line bundle of 1 is trivial,
and hence the pairing is nondegenerate. Next consider the short exact sequence of vector bundles on 1 given by the
differential of u
du
0 −→ T 1 −→ u ∗ TX −→ Q := (u ∗ TX )/ T 1 −→ 0 .
(5.1)
Let
ρ
H 0 (1 , Q ) −→ H 1 (1 , T 1 ) −→ H 1 (1 , u ∗ TX ) −→ H 1 (1 , Q )
(5.2)
be the long exact sequence of cohomologies associated with it. We have H (1 , Q ) = 0 because degree( Q ) > 0 (note
that degree(u ∗ TX ) > 0 and degree( T 1 ) = 0). The exact sequence in (5.1) does not split; for the corresponding extension
class ψ ∈ H 1 (1 , ( T 1 ) ⊗ Q ∗ ) = H 1 (1 , Q ∗ ), as observed before, there is ψ ∈ H 0 (1 , Q ) such that ψ ∪ ψ = 0. Hence
ρ in (5.2) is nonzero. This implies that ρ is surjective because dim H 1 (1 , u ∗ TX ) = 1. Hence from (5.2) we conclude that
H 1 (1 , u ∗ TX ) = 0, which proves that the cokernel of D u is zero (i.e., D u is surjective). 2
1
When X := P1 × P1 , the complex structure is genus one regular; that is because for P1 the complex structure is genus
one regular (by [10, Corollary 6.5]). Hence, the same fact holds for products of P1 .
Acknowledgements
We are very grateful to the two referees for detailed comments. The second author is grateful to Hannah Markwig and
Yoav Len for some fruitful discussions on this subject and informing us about the ongoing work [7]. We thank them for
letting us know that this question is of interest to tropical geometers.
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